Definition 1. Linear operator $L:\mathsf{H}\to\mathsf{H}$ is bounded if
\begin{equation}
\| Lu\|\le M\|u\|\qquad \forall u\in \mathsf{H};
\label{equ-L.1}
\end{equation}
the smallest constant $M$ for which it holds is called operator norm of $L$ and denoted $\|L\|$.
Definition 2. Let $L:\mathsf{H}\to\mathsf{H}$ be a bounded linear operator.
Adjoint operator $L^*$ is defined as
\begin{equation}
(Lu, v)= (u,L^*v) \qquad \forall u,v\in \mathsf{H};
\label{equ-L.2}
\end{equation}
Operator $L$ is self-adjoint if $L^*=L$.
However one needs to consider also unbounded operators. Such operators not only fail (\ref{equ-L.1}) but they are not defined everywhere.
Definition 3. Consider a linear operator $L:D(L)\to \mathsf{H}$ where $D(L)$ is a linear subset in $\mathsf{H}$ (i.e. it is a linear subspace but we do not call it this way because it is not closed) which is dense in $\mathsf{H}$ (i.e. for each $u\in \mathsf{H}$ there exists a sequence $u_n \in D(L)$ converging to $u$ in $\mathsf{H}$). Then
Operator $L$ is closed if $u_n\to u$, $Lu_n\to f$ imply that $u\in D(L)$ and $Lu=f$;
Operator $L$ is symmetric if \begin{equation}
(Lu, v)= (u,L v) \qquad \forall u,v\in D(L);
\label{equ-L.3}
\end{equation}
Symmetric operator $L$ is self-adjoint if $(L\pm i)^{-1}:\mathsf{H}\to D(L)$ exist: $(L\pm i)(L\pm i)^{-1}=I$, $(L\pm i)^{-1}(L\pm i)=I$ (identical operator)
For bounded operators "symmetric" equals "self-adjoint";
Not so for unbounded operators. F.e. $Lu=-u''$ on $(0,l)$ with
$D(L)=\{u(0)=u'(0)=u(l)=u'(l)=0\}$ is symmetric but not self-adjoint;
Self-adjoint operators have many properties which symmetric but not self-adjoint operators do not habe;
In Quantum Mechanics observables are self-adjoin operators.
Definition 4. Let us consider operator $L$ (bounded or unbounded). Then
$z\in \mathbb{C}$ belongs to the resolvent set of $L$ if $(L-z)^{-1}: \mathsf{H}\to D(L)$ exists and is a bounded operator: $(L-z)^{-1}(L-z)=I$, $(L-z)(L-z)^{-1}=I$.
$z\in \mathbb{C}$ belongs to the spectrum of $L$ if it does not belong to its resolvent set. We denote spectrum of $L$ as $\sigma (L)$.
$z$ is an eigenvalue if there exists $u\ne 0$ s.t. $(A-z)u=0$. Then $u$ is called eigenvector (or eigenfunction depending on context) and $\{u:\, (A-z)u=0\}$ is an eigenspace (corresponding to eigenvalue $z$). The dimension of the eigenspace is called a multiplicity of $z$. Eigenvalues of multiplicity $1$ are simple, eigenvalues of multiplicity $2$ are double, ... but there could be eignvalues of infinite multiplicity!
The set of all eigenvalues is called pure point spectrum;
Eigenvalues of the finite multiplicity which are isolated from the rest of the spectrum form a discrete spectrum; the rest of the spectrum is called essential spectrum.
Definition 6.
$z\in \mathbb{C}$ belongs to continuous spectrum if $z$ is not an eigenvalue and inverse operator $(L-z)^{-1}$ exists but is not a bounded operator (so its domain $D((L-z)^{-1}$ is dense).
Remark 3.
Continuous spectrum could be classified as well. The difference between absolutely continuous and singular continuous spectra will be illustrated but one can define also multiplicity of continuous spectrum as well. However one needs a Spectral Theorem to deal with these issues properly.